HSC Commerce (English Medium) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Using the truth table, verify

~(p ∨ q) ∨ (~ p ∧ q) ≡ ~ p

Prove that the following pair of statement pattern is equivalent.

p ↔ q and (p → q) ∧ (q → p)

Prove that the following pair of statement pattern is equivalent.

p → q and ~ q → ~ p and ~ p ∨ q

Prove that the following pair of statement pattern is equivalent.

~(p ∧ q) and ~p ∨ ~q

Write the dual of the following:

(p ∨ q) ∨ r

Write the dual of the following:

~(p ∨ q) ∧ [p ∨ ~ (q ∧ ~ r)]

Write the dual of the following:

p ∨ (q ∨ r) ≡ (p ∨ q) ∨ r

Write the dual of the following:

~(p ∧ q) ≡ ~ p ∨ ~ q

Write the dual statement of the following compound statement.

13 is prime number and India is a democratic country.

Write the dual statement of the following compound statement.

Karina is very good or everybody likes her.

Write the dual statement of the following compound statement.

Radha and Sushmita cannot read Urdu.

Write the dual statement of the following compound statement.

A number is a real number and the square of the number is non-negative.

Write the negation of the following statement.

All the stars are shining if it is night.

Write the negation of the following statement.

∀ n ∈ N, n + 1 > 0

Write the negation of the following statement.

∃ n ∈ N, (n² + 2) is odd number.

Write the negation of the following statement.

Some continuous functions are differentiable.

Using the rules of negation, write the negation of the following:

(p → r) ∧ q

Using the rules of negation, write the negation of the following:

~(p ∨ q) → r

Using the rules of negation, write the negation of the following:

(~p ∧ q) ∧ (~q ∨ ~r)

Write the converse, inverse, and contrapositive of the following statement.

"If it snows, then they do not drive the car"